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Chapter 1: Problem 15

Simplify the quantities using \(m(z)=z^{2}\). $$m(z+h)-m(z)$$

Short Answer

Expert verified
The expression simplifies to \(2zh + h^2\).

Step by step solution

01

Identify Function Definition

We are given the function \(m(z) = z^2\). We need to substitute this definition into the expression \(m(z+h) - m(z)\).
02

Substitute Functions into Expression

Replace \(m(z+h)\) with \((z+h)^2\) and \(m(z)\) with \(z^2\). The expression becomes \((z+h)^2 - z^2\).
03

Expand \((z+h)^2\)

Use the formula for the square of a binomial: \((z+h)^2 = z^2 + 2zh + h^2\). Substitute this back into the expression: \(z^2 + 2zh + h^2 - z^2\).
04

Simplify the Expression

Notice that \(z^2\) and \( - z^2\) cancel each other out. We are left with \(2zh + h^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Simplification
Function simplification refers to the process of making a mathematical expression more concise without changing its value. In our case, we start with the function given as \(m(z) = z^2\). The task is to simplify the expression \(m(z+h) - m(z)\).
Substitute the function definition into this expression.
  • First, replace \(m(z+h)\) with \((z+h)^2\).
  • Next, replace \(m(z)\) with \(z^2\).
This substitution transforms the expression into \((z+h)^2 - z^2\). Simplification helps in reducing complexity and focuses on essential terms. By organizing the expression neatly, we set the stage for further simplification steps, such as expanding and evaluating the difference quotient.
Binomial Expansion
Binomial expansion is a crucial method to simplify expressions involving binomials raised to a power. Let's apply it to our expression \((z+h)^2 - z^2\).
To expand \((z+h)^2\), we use the formula for the square of a binomial:
  • \((z+h)^2 = z^2 + 2zh + h^2\)
Insert this expanded form back into the original expression, resulting in \(z^2 + 2zh + h^2 - z^2\).
The expanded form shows all terms clearly, and we can easily identify terms that will cancel out. Notice the \(z^2\) and \(- z^2\), which simplify to zero. This step makes binomial expansion a powerful tool in differential calculus when simplifying terms.
Difference Quotient
The difference quotient is a fundamental concept used to compute the derivative of a function. It's defined as the ratio:\[\frac{f(x+h) - f(x)}{h}\]This quotient gives us a way to understand the function's rate of change. In our exercise, we're initially interested in \(m(z+h) - m(z)\) before considering the quotient.
  • The expression reduces to \(2zh + h^2\) after simplification.
Understanding the difference quotient's form helps bridge the simplicity of function definitions and the complexity of calculus derivatives. As \(h\) approaches zero, the difference quotient approximates the derivative, the symbolic representation of the instantaneous rate of change at any given point.

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Most popular questions from this chapter

Explain what is wrong with the statement. \(f(x)=3 x+5\) and \(g(x)=-3 x-5\) are inverse functions of each other.

Show \(f(x)=x\) is continuous everywhere.

Suppose \(f\) is an increasing function and \(g\) is a decreasing function. Give an example for \(f\) and \(g\) for which the statement is true, or say why such an example is impossible. \(f(x)-g(x)\) is decreasing for all \(x\).

For the functions in Problems \(46-53,\) do the following: (a) Make a table of values of \(f(x)\) for \(x=0.1,0.01,0.001\) \(0.0001,-0.1,-0.01,-0.001,\) and -0.0001 (b) Make a conjecture about the value of \(\lim _{x \rightarrow 0} f(x)\) (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for \(x\) near 0 such that the difference between your conjectured limit and the value of the function is less than \(0.01 .\) (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.) $$f(x)=\frac{e^{x}-1}{x}$$

If \(y=f(x)\) is a linear function, then increasing \(x\) by 1 unit changes the corresponding \(y\) by \(m\) units, where \(m\) is the slope.
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